Threshold for Steiner triple systems

Abstract

We prove that with high probability ${\mathbb{G}}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3(\mathrm{mod}6)$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.

Improved exposition. Results unchanged. 23 pages, 1 figure